Let $\mathbf{v}_0$ be a vector.  The vector $\mathbf{v}_0$ is projected onto $\begin{pmatrix} 3 \\ 1 \end{pmatrix},$ resulting in the vector $\mathbf{v}_1.$  The vector $\mathbf{v}_1$ is then projected onto $\begin{pmatrix} 1 \\ 1 \end{pmatrix},$ resulting in the vector $\mathbf{v}_2.$  Find the matrix that takes $\mathbf{v}_0$ to $\mathbf{v}_2.$
Explanation: The matrix that projects onto $\begin{pmatrix} 3 \\ 1 \end{pmatrix}$ is
\[\begin{pmatrix} \frac{9}{10} & \frac{3}{10} \\ \frac{3}{10} & \frac{1}{10} \end{pmatrix},\]and the matrix that projects onto $\begin{pmatrix} 1 \\ 1 \end{pmatrix}$ is
\[\begin{pmatrix} \frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{2} \end{pmatrix},\]so the matrix that takes $\mathbf{v}_0$ to $\mathbf{v}_2$ is
\[\begin{pmatrix} \frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{2} \end{pmatrix} \begin{pmatrix} \frac{9}{10} & \frac{3}{10} \\ \frac{3}{10} & \frac{1}{10} \end{pmatrix} = \boxed{\begin{pmatrix} \frac{3}{5} & \frac{1}{5} \\ \frac{3}{5} & \frac{1}{5} \end{pmatrix}}.\]